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I am trying to understand a text about beamforing :

Classic beamforming toy example. Single spatial stream, 2x2. With one interferer, receiver alone can null it. With two interferers, need to coordinate nulls in both receivers and transmitters. For example, the nulls for the transmitters to receivers are (1,2), (2,3), (3,1) and for the receivers to transmitters are (2,1), (3,2), and (1,3), thus using all six degrees of freedom to cancel the six interference paths. With three interferers (four user pairs), we lack sufficient degrees of freedom to cancel interference. Thus, we resort to orthogonalization of channels.

What do we mean by nulling in this case? Are interference coming from the antennas inside the 2x2 system? is the case with two interferes a 3x3 system and the next one a 4x4 system? What is a user pair, is it an antenna in the MIMO setup? What is a degree of freedom in this case and why do we have only six of them?

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  • \$\begingroup\$ Welcome to EE.SE. Could you add a little into your post explaining what "beamforming" is (for the rest of us)? You should also reference the text you mentioned so that others can get context if required. \$\endgroup\$
    – Transistor
    Commented Aug 23, 2016 at 21:05
  • \$\begingroup\$ Welcome to EE.SE. Your question is a wall of text. It's very hard to digest. Try to reduce the number of questions to 1. Try to very clearly structure your text visually – visually insert a boundary between definition and example. \$\endgroup\$
    – Marcus Müller
    Commented Aug 23, 2016 at 21:08
  • \$\begingroup\$ @NickAlexeev I don't think MIMO is accurate here. By my understanding/guess of the question, I think the OP is using a 2-dimensional array of 4 antennae sending/receiving a phase-corrected copy of the same identical signal in order to "steer the beam" of an RF transmission using the phase-induced constructive/destructive interference. -- Might be a better-place question over on the Amateur Radio stack tho. \$\endgroup\$
    – Robherc KV5ROB
    Commented Aug 23, 2016 at 21:10
  • \$\begingroup\$ @RobhercKV5ROB don't thin that the amateur radio SE would have significantly more knowledge than this SE site filled with Electrical Engineers :) The problem really is that karb22 should clearly define things, then answers might be derived really easily. The way karb22 underdefines the problem would require anyone to write a 20 page report on how to model such systems. \$\endgroup\$
    – Marcus Müller
    Commented Aug 23, 2016 at 21:21
  • \$\begingroup\$ Thanks for the quick comments. This is an excerpt from a proposal draft (printed) that my professor has given me and left me with for a month to figure it out. I just started graduate school and I'm trying learn these things on my own for now and before the semester starts. \$\endgroup\$
    – karb22
    Commented Aug 23, 2016 at 21:21

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Nulling is a RF technique for dealing with sources of interference. A boring example might be the noise coming off of a blender motor. When you null an interference source out, you identify the direction it comes from and then adjust the relative phases between the antennas such that signals from that particular direction interfere destructively, canceling out the noise.

The degrees of freedom are key because you only have so many knobs to twiddle. If you have a 2x2 system, you can adjust four phases, but what will matter for the purposes of nulling is the relationship between the phases: you have 3 relative phases that you can adjust. If you have one source of interference, its direction is well defined by an azimuth and elevation, so you have enough degrees of freedom to put a null at exactly that point.

In the case of two interference sources, you have to put 2 nulls into your directional antenna response. Each one has an azimuth and an elevation, so your problem has 4 degrees of freedom. Unfortunately, you only have 3 degrees of freedom in your antenna, so it is impossible for just the receiver to cancel out both signals. With a more advanced antenna pattern, you may be able to cancel out more.

I'm not familiar with cancellation where both transmitter and receiver cooperate to nullify multiple interference sources, so you'll need to look at the literature to better understand how that works. However, hopefully that is enough to get you started!

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